Refracting telescopes: principle and geometrical optics basics (Gauss' approximation)

Telescopes have been use for a long time to observe the sky

Actually, the observation of stars and planets is what led us to many major progress in physics... It enabled us, among other things, to understand gravity to such an extent that we can now precisely predict any planet movement. (though we still don't know what's happening at nano scales but it's another topic...)

Instruments are needed to observe celestial bodies

One of the first of its kind was the refracting telescope that relies on simple optical properties. It uses lenses.

In this article we will focus (pun intended) on thin lenses.

Diverging and converging lenses, basics of geometrical optics

if you're already familiar with geometrical optics there is no need for you to read this part that basically recaps the construction of light rays going through lenses.

Lenses have one main characteristics which is their focal length f'.

This focal distance represents the distance from the focus object (F) to the optical center (O the intersection between Δ and the lens) which happen to be the same as the distance between the focus image (F') to O.

In geometrical optics we consider light as a buch of light rays that we will represent as (usually) straight lines.

A couple of rules describe the trajectory of a light ray ing through a lens:

-a light ray ing by F will leave the lens parallel to Δ

-a light ray ing by O will not be affected by the lens

-a light ray arriving parallel to Δ will leave the lens ing by F'

While it seems pretty straight forward when considering converging lens where F is upstream from the lens and F' after, this might be a but trickier when it comes to considering diverging lenses because the position of the two focuses are flipped.

What we have to do is consider the light ray as infinite when it goes out so that one of its end can through the point it is supposed to in. This is not really clear but a schematic is worth more than a thousand words so here are the two types of lenses:

For instance the light ray coming out from B parallel to Δ that hits the diverging lens is supposed to by F' according to the rules. However F' is on the left and a light ray always keeps on going forward. So we will trace the line that s F' and the hitting point on the lens. The part of that line after the lens is the actual light ray coming out of it.

On those two exemples you can notice that the light rays are converging toward a single point. This point is called an image because if we were to put a screen at this place, it would form the image of the B point of the object (arrow) from where the light ray emitted.

'Another thing to notice is that while on our example the image was flipped with theconverging lens it wasn't with the diverging one. This actually depends on the position of the object in relation to the focus.

You can try to make the other cases if you want to "have fun".

The case of the refracting telescope

In the previous examples we considered that the light rays were coming from an object and now it is still the case except that the object is sooo far aways that we'll consider it infinitely far away.

In this case light rays all come parrallel.

However the previous rules of construction still work.

The first refracting telescope of its kind was basically composed of two converging lenses.

We're gonna call the first L_1 (with F_1 and F'_1) its focuses and the second L_2 (with F_2 and F'_2) its focuses.

Those two lenses have different focal length and are place so that F'_1=F_2.

Then, when parrallel rays hits L_1 at an angle α they will leave L_2 still parrallels but at a greater angle α'. (see the schematic)

The physical definition of magnification is M=α' /α

The optical system magnified the original image: the sky.

But how does M relate with f'_1 and f'_2?

Well the demonstration does not involve differential equation this time and is actually approachable with simple junior high geometry.

Here it is:

The approximation of tan α = α is actually valid because we are in Gauss approximation that guarantees little angles. If α is small, doing a finite expansion of tan (first term) immediately gives us tan α ≈ α.

What do we notice?

The image that will be seem through the telescope will actually be flipped! Indeed, rays come in with a negative slope and goes out with a positive one.

Galileo in fact invented another type of refracting telescope involving a converging and a diverging lens that does not flip the image (with the same expression of magnification)

Here is it's principle (the photo is also his telescope)

Is it still used nowadays?

By amateurs for sure however scientists now prefer reflecting telescopes or radar telecopes that are far more convinient to observe stars and more generally the sky.

If you want me to do an article on a specific subject I would be happy to here your suggestions!

Sources:

My physics class

https://artsandculture.google.com/asset/lunette-astronomique-galil%C3%A9e/1AF541lS6PMG0Q